Asymptotics of Resonances for Radial Potential Scattering in Hyperbolic Space Open Access

Abstract

Abstract
Asymptotics of Resonances for Radial Potential Scattering in Hyperbolic Space
It is shown that for scattering by a spherically symmetric potential with
compact support in Hⁿ⁺¹ , the resonance counting function N (r) is asymp-
totic to Crⁿ⁺¹. C is found explicitly, and is shown to depend only on the
dimension and the radius of support of the potential.

Table of Contents

1 Introduction 1

2 Estimates on the Scattering Matrix 6

2.1 Relative Scattering Theory Background . . . . . . . . . 6

2.2 Form of the Relative Scattering Matrix . . . . . . . . . 10

2.3 Scattering Matrix Estimates . . . . . . . . . . . . . . . . 13

2.3.1 Background Estimates . . . . . . . . . . . . . . . . . . . 14

2.3.2 Estimating the Exponentially Decaying Terms . . . 17

2.3.3 Estimating the Exponentially Growing Terms . . . . 20

2.3.4 Estimating the Leading Exponential Growth Term . 23

2.4 Proof of the Bound on the Scattering Matrix . . . . . 27

2.4.1 Estimating the Scattering Matrix for Small k . . . . 29

3 Asymptotics for the Number of Scattering Poles . . . . 31

Appendix 40

4.1 Scattering Determinant Estimate . . . . . . . . . . . . 40

4.2 Scattering Phase Estimate . . . . . . . . . . . . . . . . . 40

4.3 Laplace's Method . . . . . . . . . . . . . . . . . . . . . . . 44

Bibliography 48

About this Dissertation

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School	Laney Graduate School
Department	Math and Computer Science
Degree	PhD
Submission	Dissertation
Language	English
Research Field	Mathematics
Keyword	spectral theory resonances hyperbolic space radial potential scattering
Committee Chair / Thesis Advisor	Borthwick, David, Emory University
Committee Members	Hamilton, Emily Margaret, Emory University Yang, Shanshuang, Emory University

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